Linear connections for reproducing kernels on vector bundles

Abstract

Abstract. We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.

Geometric Perspectives on Reproducing Kernels

“…The proof is an application of Proposition 1 along with the universal Theorem 3 from which ı K ı r K D r S 0 Q ı T . K /, where r S 0 is the covariant derivative for the universal connection on … H;S 0 [6]. In the setting of Theorems 4 and 6, it can be shown that the covariant derivative r K is a Chern covariant derivative compatible both with the complex structure of the dual bundle … W D !…”

“…In particular, …W D ! Z with an admissible reproducing kernel K is isomorphic to the pullback of the tautological bundle … H K through K ; see [6,Th. 3.11].…”

“…The above functor A is a unique subject to a certain factorizing property that involves categories of Hilbert spaces, of Hermitian vector bundles, and of linear connections. Details can be found in [6].…”

Berezin Symbols on Lie Groups

“…This problem is also related to some representations of infinite-dimensional Lie groups that occur in the study of magnetic fields (see [BB11a] and [BB12]). Let us also mention that linear differential operators associated to reproducing kernels have been earlier used in the literature (see for instance [BG14]).…”

Reproducing Kernels and Positivity of Vector Bundles in Infinite Dimensions

“…Maybe we should also mention at this point that the interaction between complex geometry and operator theory has already surfaced in the literature, as for instance in the famous theory of the Cowen-Douglas operators ( [CD78]), without to emphasize particularly the idea of positivity. As regards the specific topic of the present paper, we will discuss the relationship between the Griffiths positivity of holomorphic vector bundles ( [Gr69], [GH78]) and the reproducing kernels on infinite-dimensional vector bundles that we have studied recently ( [BG08], [BG09], [BG11], [BG13]).…”